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Proofgold Asset
asset id
22cfca5bb314ac046d24960c9e7a2d9ed5a03bad6f62759f721a6e3d2ee64ecf
asset hash
859a157319cd16414fc83eb876e03cd43b9dc37a015ae2f19fe375c6b8358be4
bday / block
35443
tx
6caa7..
preasset
doc published by
Pr4zB..
Param
4402e..
:
ι
→
(
ι
→
ι
→
ο
) →
ο
Param
cf2df..
:
ι
→
(
ι
→
ι
→
ο
) →
ο
Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Param
setminus
setminus
:
ι
→
ι
→
ι
Param
Sing
Sing
:
ι
→
ι
Definition
False
False
:=
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Definition
8b6ad..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 .
∀ x5 : ο .
(
(
x1
=
x2
⟶
∀ x6 : ο .
x6
)
⟶
(
x1
=
x3
⟶
∀ x6 : ο .
x6
)
⟶
(
x2
=
x3
⟶
∀ x6 : ο .
x6
)
⟶
(
x1
=
x4
⟶
∀ x6 : ο .
x6
)
⟶
(
x2
=
x4
⟶
∀ x6 : ο .
x6
)
⟶
(
x3
=
x4
⟶
∀ x6 : ο .
x6
)
⟶
not
(
x0
x1
x2
)
⟶
not
(
x0
x1
x3
)
⟶
not
(
x0
x2
x3
)
⟶
not
(
x0
x1
x4
)
⟶
not
(
x0
x2
x4
)
⟶
not
(
x0
x3
x4
)
⟶
x5
)
⟶
x5
Definition
c5756..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 .
∀ x6 : ο .
(
8b6ad..
x0
x1
x2
x3
x4
⟶
(
x1
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x2
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x3
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x4
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
not
(
x0
x1
x5
)
⟶
not
(
x0
x2
x5
)
⟶
x0
x3
x5
⟶
x0
x4
x5
⟶
x6
)
⟶
x6
Definition
f8709..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 .
∀ x7 : ο .
(
c5756..
x0
x1
x2
x3
x4
x5
⟶
(
x1
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x2
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x3
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x4
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x5
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
not
(
x0
x1
x6
)
⟶
x0
x2
x6
⟶
x0
x3
x6
⟶
x0
x4
x6
⟶
not
(
x0
x5
x6
)
⟶
x7
)
⟶
x7
Definition
e8ae3..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
f8709..
x0
x1
x2
x3
x4
x5
x6
⟶
(
x1
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x2
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x3
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x4
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x5
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x6
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
x0
x1
x7
⟶
not
(
x0
x2
x7
)
⟶
x0
x3
x7
⟶
x0
x4
x7
⟶
not
(
x0
x5
x7
)
⟶
not
(
x0
x6
x7
)
⟶
x8
)
⟶
x8
Definition
fa72d..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
e8ae3..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
not
(
x0
x1
x8
)
⟶
not
(
x0
x2
x8
)
⟶
not
(
x0
x3
x8
)
⟶
x0
x4
x8
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
06d7e..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 x9 .
∀ x10 : ο .
(
fa72d..
x0
x1
x2
x3
x4
x5
x6
x7
x8
⟶
(
x1
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x2
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x3
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x4
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x5
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x6
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x7
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x8
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
x0
x1
x9
⟶
x0
x2
x9
⟶
not
(
x0
x3
x9
)
⟶
not
(
x0
x4
x9
)
⟶
x0
x5
x9
⟶
not
(
x0
x6
x9
)
⟶
not
(
x0
x7
x9
)
⟶
x0
x8
x9
⟶
x10
)
⟶
x10
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Definition
nIn
nIn
:=
λ x0 x1 .
not
(
x0
∈
x1
)
Known
setminusE
setminusE
:
∀ x0 x1 x2 .
x2
∈
setminus
x0
x1
⟶
and
(
x2
∈
x0
)
(
nIn
x2
x1
)
Definition
or
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Known
xm
xm
:
∀ x0 : ο .
or
x0
(
not
x0
)
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
53a3c..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
not
(
x1
x2
x3
)
⟶
not
(
x1
x3
x2
)
)
⟶
cf2df..
x0
x1
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
(
x2
=
x3
⟶
∀ x7 : ο .
x7
)
⟶
(
x2
=
x4
⟶
∀ x7 : ο .
x7
)
⟶
(
x3
=
x4
⟶
∀ x7 : ο .
x7
)
⟶
(
x2
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x3
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x4
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x2
=
x6
⟶
∀ x7 : ο .
x7
)
⟶
(
x3
=
x6
⟶
∀ x7 : ο .
x7
)
⟶
(
x4
=
x6
⟶
∀ x7 : ο .
x7
)
⟶
(
x5
=
x6
⟶
∀ x7 : ο .
x7
)
⟶
not
(
x1
x2
x3
)
⟶
not
(
x1
x2
x4
)
⟶
not
(
x1
x3
x4
)
⟶
not
(
x1
x2
x5
)
⟶
not
(
x1
x3
x5
)
⟶
not
(
x1
x4
x5
)
⟶
not
(
x1
x2
x6
)
⟶
not
(
x1
x3
x6
)
⟶
not
(
x1
x4
x6
)
⟶
not
(
x1
x5
x6
)
⟶
False
Known
61345..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
4402e..
x0
x1
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
(
x2
=
x3
⟶
∀ x5 : ο .
x5
)
⟶
(
x2
=
x4
⟶
∀ x5 : ο .
x5
)
⟶
(
x3
=
x4
⟶
∀ x5 : ο .
x5
)
⟶
x1
x2
x3
⟶
x1
x2
x4
⟶
x1
x3
x4
⟶
False
Known
Subq_tra
Subq_tra
:
∀ x0 x1 x2 .
x0
⊆
x1
⟶
x1
⊆
x2
⟶
x0
⊆
x2
Known
setminus_Subq
setminus_Subq
:
∀ x0 x1 .
setminus
x0
x1
⊆
x0
Known
SingI
SingI
:
∀ x0 .
x0
∈
Sing
x0
Theorem
a1564..
:
∀ x0 x1 .
∀ x2 :
ι →
ι → ο
.
(
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
x2
x3
x4
⟶
x2
x4
x3
)
⟶
4402e..
x1
x2
⟶
cf2df..
x1
x2
⟶
∀ x3 .
x3
∈
x1
⟶
x0
⊆
setminus
x1
(
Sing
x3
)
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
∀ x9 .
x9
∈
x0
⟶
∀ x10 .
x10
∈
x0
⟶
∀ x11 .
x11
∈
x0
⟶
∀ x12 .
x12
∈
x0
⟶
06d7e..
x2
x4
x5
x6
x7
x8
x9
x10
x11
x12
⟶
∀ x13 : ο .
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
not
(
x2
x11
x3
)
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
not
(
x2
x11
x3
)
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
not
(
x2
x11
x3
)
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
not
(
x2
x11
x3
)
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
not
(
x2
x11
x3
)
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x2
x11
x3
⟶
not
(
x2
x12
x3
)
⟶
x13
)
⟶
x13
(proof)